Having used randomForest in R to produce a fairly successful classifier is there any way to emphasise sensitivity over specificity, for example, if the cost of missing a disease is much greater than diagnosing a false positive?
I don't actually know R package, but if you were able to decrease the cost of False Positives it should work. This is also the rationale of MetaCost by Domingos, that is implemented in WEKA. However, increasing the sensitivity you are going to decrease the specificity, naturally. From the information retrieval point of view, as long as you increase the recall the precision will decrease.
Because Random Forest use Decision Trees as base classifiers and they can output probabilities, you can decrease the cut-off that enable a tree to classify a record as positive. This will make you Random Forest more sensitive but less precise.